Solve for $x$ : $ 2|x + 7| - 4 = -1|x + 7| + 5 $
Solution: Add $ {1|x + 7|} $ to both sides: $ \begin{eqnarray} 2|x + 7| - 4 &=& -1|x + 7| + 5 \\ \\ { + 1|x + 7|} && { + 1|x + 7|} \\ \\ 3|x + 7| - 4 &=& 5 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 3|x + 7| - 4 &=& 5 \\ \\ { + 4} &=& { + 4} \\ \\ 3|x + 7| &=& 9 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x + 7|} {{3}} = \dfrac{9} {{3}} $ Simplify: $ |x + 7| = 3$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 7 = -3 $ or $ x + 7 = 3 $ Solve for the solution where $x + 7$ is negative: $ x + 7 = -3 $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& -3 \\ \\ {- 7} && {- 7} \\ \\ x &=& -3 - 7 \end{eqnarray} $ $ x = -10 $ Then calculate the solution where $x + 7$ is positive: $ x + 7 = 3 $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& 3 \\ \\ {- 7} && {- 7} \\ \\ x &=& 3 - 7 \end{eqnarray} $ $ x = -4 $ Thus, the correct answer is $x = -10 $ or $x = -4 $.